In this paper, we introduce operators that are represented by upper triangular \(2\times 2\) block matrices whose entries satisfy some algebraic constraints. We call them Brownian-type operators of class \({\mathcal {Q}},\) briefly operators of class \({\mathcal {Q}}.\) These operators emerged from the study of Brownian isometries performed by Agler and Stankus via detailed analysis of the time shift

In this paper, we are concerned with the following question: if the tensor product of finitely generated modules M and N over a local complete intersection domain is maximal Cohen–Macaulay, then must M or N be a maximal Cohen–Macaulay? Celebrated results of Auslander, Lichtenbaum, and Huneke and Wiegand yield affirmative answers to the question when the ring considered has codimension zero or one,

A coprime commutator in a profinite group G is an element of the form [x, y], where x and y have coprime order and an anti-coprime commutator is a commutator [x, y] such that the orders of x and y are divisible by the same primes. In the present paper, we establish that a profinite group G is finite-by-pronilpotent if the cardinality of the set of coprime commutators in G is less than \(2^{\aleph _0}\)

In this article, we characterize all eigenfunctions of the Laplacian on homogeneous trees, which are the Poisson transform of \(L^p\) functions defined on the boundary. Using the duality argument, we also proved the restriction theorem for the Helgason–Fourier transforms on a homogeneous tree.

Symmetry plays a basic role in variational problems (settled, e.g., in \({\mathbb {R}}^{n}\) or in a more general manifold), for example, to deal with the lack of compactness which naturally appears when the problem is invariant under the action of a noncompact group. In \({\mathbb {R}}^n\), a compactness result for invariant functions with respect to a subgroup G of \(\mathrm {O}(n)\) has been proved

In this paper, we shall be concerned with evaluation of Hewitt–Stromberg measure of Cartesian product sets by means of the measure of their components. This is done by the construction of new multifractal measures, on the Euclidean space \({\mathbb {R}}^n\), in a similar manner to Hewitt–Stromberg measures but using the class of all n-dimensional half-open binary cubes of covering sets in the definition

We prove an existence result for the Poisson equation on non-compact Riemannian manifolds satisfying weighted Poincaré inequalities outside compact sets. Our result applies to a large class of manifolds including, for instance, all non-parabolic manifolds with minimal positive Green’s function vanishing at infinity. On the source function, we assume a sharp pointwise decay depending on the weight appearing

We study the Hermitian curvature flow of locally homogeneous non-Kähler metrics on compact complex surfaces. In particular, we characterize the long-time behavior of the solutions to the flow. We also provide the first example of a compact complex non-Kähler manifold admitting a finite time singularity for the Hermitian curvature flow. Finally, we compute the Gromov–Hausdorff limit of immortal solutions

We are interested in differential forms on mixed-dimensional geometries, in the sense of a domain containing sets of d-dimensional manifolds, structured hierarchically so that each d-dimensional manifold is contained in the boundary of one or more \(d + 1\)-dimensional manifolds. On any given d-dimensional manifold, we then consider differential operators tangent to the manifold as well as discrete

We give a notion of “combinatorial proximity” among strongly stable ideals in a given polynomial ring with a fixed Hilbert polynomial. We show that this notion guarantees “geometric proximity” of the corresponding points in the Hilbert scheme. We define a graph whose vertices correspond to strongly stable ideals and whose edges correspond to pairs of adjacent ideals. Every term order induces an orientation

We study the space of derivations for some finite-dimensional evolution algebras, depending on the twin partition of an associated directed graph. For evolution algebras with a twin-free associated graph, we prove that the space of derivations is zero. For the remaining families of evolution algebras, we obtain sufficient conditions under which the study of such a space can be simplified. We accomplish

A remarkable result of Thompson states that a finite group is soluble if and only if all its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory of groups, aiming for global properties of groups from local properties of two-generated (or more generally, n-generated) subgroups. We contribute an extension

We analyze some properties of a class of multiexponential maps appearing naturally in the geometric analysis of Carnot groups. We will see that such maps can be useful in at least two interesting problems: first, in relation to the analysis of some regularity properties of horizontally convex sets. Then, we will show that our multiexponential maps can be used to prove the Pansu differentiability of

Let V be a vector bundle over a smooth curve C. In this paper, we study twisted Brill–Noether loci parametrising stable bundles E of rank n and degree e with the property that \(h^0 (C, V \otimes E) \ge k\). We prove that, under conditions similar to those of Teixidor i Bigas and of Mercat, the Brill–Noether loci are nonempty and in many cases have a component which is generically smooth and of the

In this paper we develop fundamental theories for a scalar first-order hyperbolic partial differential equation with two internal variables which models single-species population dynamics with two physiological structures such as age–age, age–maturation, age–size, and age–stage. Classical techniques of treating structured models with a single internal variable are generalized to study the double physiologically

Consideration in this paper is three-dimensional capillary–gravity water flows governed by the geophysical water wave equations with all the Coriolis terms being retained. It is proved that the merely possible flow exhibiting a constant vorticity vector captures vanishing vertical velocity, constant horizontal velocity and flat free surface.

We consider the scaling of the optimal constant in Korn’s first inequality for elliptic and parabolic shells which was first given by Grabovsky and Harutyunyan with hints coming from the test functions constructed by Tovstik and Smirnov on the level of formal asymptotic expansions. Here, we employ the Bochner technique in Remannian geometry to remove the assumption that the middle surface of the shell

We prove ultradifferentiable Chevelley restriction theorems for a wide range of ultradifferentiable classes. As a special case we find that isotropic functions, i.e., functions defined on the vector space of real symmetric matrices invariant under the action of the special orthogonal group by conjugation, possess some ultradifferentiable regularity if and only if their restriction to diagonal matrices

We show the existence of non-Einstein homogeneous critical metrics for any quadratic curvature functional in dimension three.

A C0-Finsler structure is a continuous function \(F:TM \rightarrow [0,\infty )\) defined on the tangent bundle of a differentiable manifold M such that its restriction to each tangent space is an asymmetric norm. We use the convolution of F with the standard mollifier in order to construct a mollifier smoothing of F, which is a one-parameter family of Finsler structures \(F_\varepsilon\) that converges

We prove the existence of nonnegative variational solutions to the obstacle problem associated with the degenerate doubly nonlinear equation $$\begin{aligned} \partial _t b(u) - {{\,\mathrm{div}\,}}(Df(Du)) = 0, \end{aligned}$$ where the nonlinearity \(b :\mathbb {R}_{\ge 0} \rightarrow \mathbb {R}_{\ge 0}\) is increasing, piecewise \(C^1\) and satisfies a polynomial growth condition. The prototype

We consider the semi-classical limit of the quantum evolution of Gaussian coherent states whenever the Hamiltonian H is given, as sum of quadratic forms, by \( H= -\frac{{\hbar ^{2}}}{2m}\,\frac{d^{2}\,}{dx^{2}}\,\dot{+}\,\alpha \delta _{0}\), with \(\alpha \in \mathbb R\) and \(\delta _{0}\) the Dirac delta-distribution at \(x=0\). We show that the quantum evolution can be approximated, uniformly

For a complete symplectic manifold \(M^{2n}\), we define the \(L^{2}\)-hard Lefschetz property on \(M^{2n}\). We also prove that the complete symplectic manifold \(M^{2n}\) satisfies \(L^{2}\)-hard Lefschetz property if and only if every class of \(L^{2}\)-harmonic forms contains a \(L^{2}\) symplectic harmonic form. As an application, we get if \(M^{2n}\) is a closed symplectic parabolic manifold

It has been known for some time that there exist 5 essentially different real forms of the complex affine Kac–Moody algebra of type \(A_2^{(2)}\) and that one can associate 4 of these real forms with certain classes of “integrable surfaces,” such as minimal Lagrangian surfaces in \(\mathbb {CP}^2\) and \(\mathbb {CH}^2\), as well as definite and indefinite affine spheres in \({\mathbb {R}}^3\). In

We study existence and uniqueness of nonnegative solutions to a problem which is modeled by $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta _p u = u^{-\theta }|\nabla u|^p + fu^{-\gamma }& \text {in}\, \Omega , \\ u=0 & \text {on}\ \partial \Omega , \end{array}\right. } \end{aligned}$$ where \(\Omega\) is an open bounded subset of \({\mathbb {R}}^N\) (\(N\ge 2\)), \(\Delta _p\) is

In this work, we establish the existence of nonzero solutions for a class of quasilinear elliptic equations involving indefinite nonlinearities with exponential critical growth of Trudinger–Moser type. Our proofs rely on variational arguments in a Orlicz–Sobolev space with a version of the Trudinger–Moser inequality.

In this paper, we study balanced metrics and Berezin quantization on a class of Hartogs domains defined by \(\varOmega _n=\{(z_1,\ldots ,z_n)\in {\mathbb {C}}^n:\vert z_1\vert<\vert z_2\vert<\cdots<\vert z_n\vert <1\}\) which generalize the so-called classical Hartogs triangle. We introduce a Kähler metric \(g(\nu )\) associated with the Kähler potential \(\varPhi _n(z):=-\sum _{k=1}^{n-1}\nu _k\ln

Motivated by the Hilbert-space model for quantum mechanics, we define a pre-Hilbert space logic to be a pair \((S,{\mathscr {L}})\), where S is a pre-Hilbert space and \({\mathscr {L}}\) is an orthocomplemented poset of orthogonally closed linear subspaces of S, closed w.r.t. finite-dimensional perturbations (i.e., if \(M\in {\mathscr {L}}\) and F is a finite-dimensional linear subspace of S, then

Fueter’s theorem (1934) asserts that every holomorphic intrinsic function of one complex variable induces an axial quaternionic monogenic function. Sce (Atti Accad Naz Lincei Rend Cl Sci Fis Mat Nat 23:220–225, 1957) generalizes Fueter’s theorem to the Euclidean spaces \({{\mathbb {R}}}^{n+1}\) for n being odd positive integers. By using pointwise differential computation he asserted that every holomorphic

The X-rank of a point p in projective space is the minimal number of points of an algebraic variety X whose linear span contains p. This notion is naturally submultiplicative under tensor product. We study geometric conditions that guarantee strict submultiplicativity. We prove that in the case of points of rank two, strict submultiplicativity is entirely characterized in terms of the trisecant lines

We construct an unbounded strictly pseudoconvex Kobayashi hyperbolic and complete domain in \({\mathbb {C}}^2\), which also possesses complete Bergman metric, but has no nonconstant bounded holomorphic functions.

This paper is a contribution to frame theory. Frames in a Hilbert space are generalizations of orthonormal bases. In particular, Gabor frames of \(L^2(\mathbb {R})\), which are made of translations and modulations of one or more windows, are often used in applications. More precisely, the paper deals with a question posed in the last years by Christensen and Hasannasab about the existence of overcomplete

This paper establishes a connection between a problem in Potential Theory and Mathematical Physics, arranging points so as to minimize an energy functional, and a problem in Combinatorics and Number Theory, constructing ’well-distributed’ sequences of points on [0, 1). Let \(f:[0,1] \rightarrow {\mathbb {R}}\) be (1) symmetric \(f(x) = f(1-x)\), (2) twice differentiable on (0, 1), and (3) such that

We assume to have information about the generating properties of the subsets of a finite group G. In particular, we consider the two following situations. We know, for every subset X of G, whether X is a generating set of G. We know the graph whose vertices are the subsets of G and in which there is an edge connecting X and Y if and only if \(X\cup Y\) is a generating set of G. We discuss how this

We extended the study of the linear fractional self maps (e.g., by Cowen–MacCluer and Bisi–Bracci on the unit balls) to a much more general class of domains, called generalized type I domains, which includes in particular the classical bounded symmetric domains of type I and the generalized balls. Since the linear fractional maps on the unit balls are simply the restrictions of the linear maps of the

We prove that if the elliptic problem \(-\Delta u+b(x)|\nabla u|=c(x)u\) with \(c\ge 0\) has a positive supersolution in a domain \(\varOmega \) of \( {\mathrm {R}}^{N\ge 3}\), then c, b must satisfy the inequality $$\begin{aligned} \sqrt{ \int _\varOmega c\phi ^2}\le \sqrt{ \int _\varOmega | \nabla \phi |^2}+\sqrt{ \int _\varOmega \frac{b^2}{4}\phi ^2},\quad \phi \in C_c^\infty (\varOmega ). \end{aligned}$$

We consider here three-dimensional water flows governed by the geophysical water wave equations exhibiting full Coriolis and centripetal terms. More precisely, assuming a constant vorticity vector, we derive a family of explicit solutions, in Eulerian coordinates, to the above-mentioned equations and their boundary conditions. These solutions are the only ones under the assumption of constant vorticity

We give necessary and sufficient conditions for the existence of entire solutions bounded or large of the equation \({\mathcal {L}}u - p\psi (u) =0\), where \({\mathcal {L}}\) is either the Laplace operator on \({\mathbb {R}} ^d\), \(d\ge 3\) or the Laplace–Beltrami operator on the harmonic NA group and p is a function whose oscillation tends to zero at infinity at a specified rate. The results apply

This paper studies the boundary behaviour of \(\lambda \)-polyharmonic functions for the simple random walk operator on a regular tree, where \(\lambda \) is complex and \(|\lambda |> \rho \), the \(\ell ^2\)-spectral radius of the random walk. In particular, subject to normalisation by spherical, resp. polyspherical functions, Dirichlet and Riquier problems at infinity are solved, and a non-tangential

In this paper, we study the supercritical problem \((P_\varepsilon )\): \( -\triangle u=K|u|^{4/(n-2)+\varepsilon }u~ \text{ in } \Omega , ~ u=0 \text{ on } \partial \Omega ,\) where \(\Omega \) is a smooth bounded domain in \(\mathbb {R}^n\), \(n=3,4\), \(\varepsilon \) is a positive real parameter and K is a \(C^4\) positive function on \({\overline{\Omega }}\). Following the ideas of Bahri et al

We completely classify Levi flat CR structures (that is, CR structures with vanishing Levi form) on three-dimensional real Lie algebras, in terms of their algebraic and almost contact properties.

This paper is devoted to the new shallow-water model (also called the modified Camassa–Holm–Novikov equation) with cubic nonlinearity, which admits the single peaked solitons and multi-peakon solutions, and includes both the Fokas–Olver–Rosenau–Qiao equation and the Novikov equation as two special cases. It is shown that the Cauchy problem of the modified Camassa–Holm–Novikov equation for the periodic

We deal with a notion of weak binormal and weak principal normal for non-smooth curves of the Euclidean space with finite total curvature and total absolute torsion. By means of piecewise linear methods, we first introduce the analogous notion for polygonal curves, where the polarity property is exploited, and then make use of a density argument. Both our weak binormal and normal are rectifiable curves

A complete classification of left-invariant closed \(G_2\)-structures on Lie groups which are extremally Ricci pinched (i.e., \( {d}\tau = \tfrac{1}{6}|\tau |^2\varphi + \tfrac{1}{6}*(\tau \wedge \tau )\)), up to equivalence and scaling, is obtained. There are five of them, they are defined on five different completely solvable Lie groups and the \(G_2\)-structure is exact in all cases except one,

In this note, we deal with compact generalized (λ, n + m)-Einstein manifolds with nonempty boundary. More precisely, we extend some known results about compact static manifolds for (λ, n + m)-Einstein manifolds, giving topological classifications for its boundary. As a consequence, we obtain some characterizations of the round hemispheres.

If a Hilbert geometry of twice differentiable boundary has two quadratic infinitesimal spheres, then the Hilbert geometry is a Cayley–Klein model of the hyperbolic geometry.

The aim of this paper is to show how the number of conjugacy classes appearing in the product of classes affect the structure of a finite group. The aim of this paper was to show several results about solvability concerning the case in which the power of a conjugacy class is a union of one or two conjugacy classes. Moreover, we show that the above conditions can be determined through the character

We establish a local Lipschitz regularity result of solutions to the Cauchy–Dirichlet problem associated with evolutionary partial differential equations $$\begin{aligned} \left\{ \begin{array}{ll} \partial _t u - {{\,\mathrm{div}\,}}Df(\nabla u) = 0, &{} \quad \text{ in } \, \Omega _T,\\ u=u_0, &{} \quad \text{ on } \, \partial _{{\mathcal {P}}}\, \Omega _T. \end{array} \right. \end{aligned}$$ We

In this paper, we prove a general stability result for higher-order geometric flows on the circle, which basically states that if the initial condition is close to a round circle, the curve evolves smoothly and exponentially fast towards a circle (possibly not the one it started close to), and we improve on known convergence rates (which we believe are almost sharp). The polyharmonic flow is an instance

We construct the tangential k-Cauchy–Fueter complexes on the right quaternionic Heisenberg group, as the quaternionic counterpart of \(\overline{\partial }_b\)-complex on the Heisenberg group in the theory of several complex variables. We can use the \(L^2\) estimate to solve the nonhomogeneous tangential k-Cauchy–Fueter equation under the compatibility condition over this group modulo a lattice. This

We prove continuous dependence on initial data for a backward parabolic operator whose leading coefficients are Osgood continuous in time. This result fills the gap between uniqueness and continuity results obtained so far.

An instanton (E, D) on a (pseudo-)hyperkähler manifold M is a vector bundle E associated with a principal G-bundle with a connection D whose curvature is pointwise invariant under the quaternionic structures of \(T_x M,~x\in M\), and thus satisfies the Yang–Mills equations. Revisiting a construction of solutions, we prove a local bijection between gauge equivalence classes of instantons on M and equivalence

We consider the physical configuration of a container which holds a finite number of movable solid objects and three immiscible fluids. Each fluid volume is prescribed, and the container is completely filled. The configuration is modeled using sets of finite perimeter, and the energy of the configuration is given using the theory of functions of bounded variation. We show that there exists an admissible

Motivated by the large amount of results obtained for minimal and positive constant mean curvature surfaces in several ambient spaces, the aim of this paper is to obtain half-space theorems for properly immersed surfaces in \({\mathbb {R}}^3\) whose mean curvature is given as a prescribed function of its Gauss map. In order to achieve this purpose, we will study the behavior at infinity of a one-parameter

In the present paper, we consider boundary value problems on the real half-line \({\varLambda }:= [0,\infty )\) of the following form $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \Big ({\varPhi }\big (a(t,x(t))\,x'(t)\big )\Big )' = f(t,x(t),x'(t)) \quad \text {a.e.}\,\text {on} \, {\varLambda }, \\ \,\,x(0) = \nu _1,\quad x(\infty ) = \nu _2, \end{array}\right. } \end{aligned}$$ where

We prove the existence of infinitely many sign-changing radial solutions for a p-sub-super critical p-Laplacian Dirichlet problem in a ball. We consider an equation defined by the p-Laplacian operator perturbed by a nonlinearity g(u) that is p-subcritical at \(+\,\infty \) and p-supercritical at \(-\,\infty \). Our results extend those in Castro et al. (Electron. J. Differ. Equ. 2007(111):1–10, 2007)

We give conditions in order to approximate locally uniformly holomorphic covering mappings of the unit ball of \({{\mathbb {C}}^{n}}\) with respect to an arbitrary norm, with entire holomorphic covering mappings. The results rely on a generalization of the Loewner theory for covering mappings which we develop in the paper.

We study the field of moduli of singular K3 surfaces. We discuss both the field of moduli over the CM field and over \({{\mathbb {Q}}}\). As a by-product, we show non-finiteness of isomorphism classes of singular K3 surfaces with field of moduli of bounded degree. Finally, we provide an explicit description of the field of moduli in terms of the 2-torsion of the Galois group of the ring class field

A commutative ring R is said to be endo-Noetherian if the chain of annihilators \({ ann}(a_{1})\subseteq \)\({ ann}(a_{2})\subseteq \cdots \) stabilizes for each sequence \((a_{k})_{k}\) of elements of R (Ndiaye and Gueye in Int J Appl Math 86:871–881, 2013). In this paper, we give equivalent conditions for the power series (resp., polynomial) rings over an Armendariz ring to be endo-Noetherian. We

We prove that if a solution of the time-dependent Schrödinger equation on an homogeneous tree with bounded potential decays fast at two distinct times then the solution is trivial. For the free Schrödinger operator, we use the spectral theory of the Laplacian and complex analysis and obtain a characterization of the initial conditions that lead to a sharp decay at any time. We then adapt the real variable